Combinatorial optimization - a field of research addressing problems that feature strongly in a wealth of scientific and industrial contexts - has been identified as one of the core potential fields of applicability of quantum computers. It is still unclear, however, to what extent quantum algorithms can actually outperform classical algorithms for this type of problems. In this work, by resorting to computational learning theory and cryptographic notions, we prove that quantum computers feature an in-principle super-polynomial advantage over classical computers in approximating solutions to combinatorial optimization problems. Specifically, building on seminal work by Kearns and Valiant and introducing a new reduction, we identify special types of problems that are hard for classical computers to approximate up to polynomial factors. At the same time, we give a quantum algorithm that can efficiently approximate the optimal solution within a polynomial factor. The core of the quantum advantage discovered in this work is ultimately borrowed from Shor's quantum algorithm for factoring. Concretely, we prove a super-polynomial advantage for approximating special instances of the so-called integer programming problem. In doing so, we provide an explicit end-to-end construction for advantage bearing instances. This result shows that quantum devices have, in principle, the power to approximate combinatorial optimization solutions beyond the reach of classical efficient algorithms. Our results also give clear guidance on how to construct such advantage-bearing problem instances.

翻译：组合优化——这一研究领域涉及在科学与工业场景中大量出现的问题——已被确认为量子计算机最核心的潜在应用领域之一。然而，对于此类问题，量子算法究竟能在何种程度上超越经典算法仍不明确。本研究通过借助计算学习理论与密码学概念，证明了量子计算机在近似求解组合优化问题时相比经典计算机具有原理性的超多项式优势。具体而言，基于Kearns与Valiant的开创性工作并引入新的归约方法，我们识别出经典计算机难以在多项式因子内近似求解的特殊问题类型。同时，我们给出了一种能在多项式因子内高效近似最优解的量子算法。本工作发现的量子优势核心实质上源于Shor的量子分解算法。具体地，我们证明了在近似求解所谓整数规划问题的特殊实例时存在超多项式优势。通过这一证明，我们提供了具有优势承载实例的显式端到端构造。该结果表明量子设备在原理上具备超越经典高效算法近似求解组合优化方案的能力。我们的研究也为如何构造此类优势承载问题实例提供了明确指引。